Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C52 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×#1043 |
| commutator, socle | C2×C52 | C23×C4 |
| Fitting | C52×#41 | C4 |
| prime radical | #41×C5⋊D5 | C2 |
| cosocle | #1039 | C24 |
| hypercenter | #41 | #1043 |
| hypocenter | C52 | #568 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ #41 | |
1032 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^6,5^2|a^2=e,d^2=f,e^2=f,d^a=df,g^a=g^2,h^a=h^2,c^b=cf,g^e=g^4,h^e=h^494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 10104);